1. Introduction

The realization of photon non-classical state [1,2] is a research hotspot of quantum optics and quantum integration technology, which is of great significance to the development and application of optical quantum field in the future. In particular, the development of single photon source technology is the main technical source to promote the development of quantum communication [3], quantum metrology [4] and quantum information technology [5,6], and the single photon source has become the ultimate goal of its product definition based on photon architecture. To this end, the study of the single-photon sources has become a conspicuous focus in quantum physics. The key of single photon source lies in the strong photon antibunching and the sub-Poisson statistics, which requires the system with strong optical nonlinearity to realize the photon blockade (PB) [7–10].

There are two main physical mechanisms for the realization of PB effect: One is conventional phonon blockade (CPB), which is induced by the splitting of the energy levels. The other is unconventional photon blockade (UPB), which is activated by the quantum destructive interference between different paths. The CPB was first observed in an optical cavity coupled to a single trapped atom [11]. After that, CPB effects were observed in different systems by several experimental groups, including the quantum dots in the photonic crystal system [12] and the quantum electrodynamics system in the circuit cavity [13,14]. At the same time, CPB effect has many different applications [15–22]. The physical mechanism of UPB is discovered by Liew and Savona, which dramatically increases the development of this research field [23]. In theory, the UPB effect can be implemented in many systems, such as, optomechanical systems [24–27], weak nonlinear photon molecules modes [28], second-order nonlinear systems [22,29–32], optical cavities with a quantum dot [33], and bimodal coupled polaritonic cavities [34]. Recently, it has been shown that the two mechanisms are connected [35,36] and they can even arise simultaneously [37,38].

In this paper, we study the UPB in a three-wave-mixing system with a non-degenerate parametric amplification. In order to obtain the optimal analytic conditions of UPB, we truncate the wave function in an effective space by retaining the Fock-state basis in the interference path and neglecting the irrelevant Fock-state basis. The numerical results are calculated by solving the master equation in a truncated Hilbert space, which agree with the optimal analytic conditions. The results indicate that the strong photon antibunching can be obtained in the high-frequency mode of the three wave mixing. Our scheme provides an approach for the UPB in a complex quantum system.

The manuscript is organized as follows: In Sec. 2, we introduce the physical model. In Sec. 3, we illustrate analytical results and physical mechanism of UPB. In Sec. 4, we show the comparison of numerical and analytical solutions for the UPB. Conclusions are given in Sec. 5.

2. Physics model

The three-wave mixing system exists a high-frequency mode $a$ with frequency $\omega _a$ and two-low-frequency modes $b$ and $c$ with frequencies $\omega _b$ and $\omega _c$, respectively. Here, the three-wave mixing mediates the convert a photon with high frequency into two photons with different low frequencies. The Hamiltonian of this system can be written as [39,40]

The Hamiltonian with non-degenerate parametric amplification and weak driving field can be expressed as:

In order to achieve photon blockade, weak driving is necessary, which is similar to the laser controlled Rydberg atomic process [41–45]. Here $F_a$ represents the weak drive intensity of mode $a$ with the frequency $\omega _{la}$, and $E$ represents the non-degenerate parametric amplification coefficient [46–49] of mode $b$ and $c$ with frequencies $\omega _{lb}$ and $\omega _{lc}$.

For convenience, we would like to turn to a rotation framework subject to the operator $\hat {U}(t)=e^{i t(\omega _{la}\hat {a}^{\dagger }\hat {a}+\omega _{lb}\hat {b}^{\dagger }\hat {b}+\omega _{lc}\hat {c}^{\dagger }\hat {c})}$, which leads to an effective Hamiltonian $\hat {H}_{eff}=\hat {U}\hat {H}\hat {U}^{\dagger }-i \hat {U}dU^{\dagger }/dt$ as

3. Analytical results

The Fock-state basis of the system is denoted by $|m,n,p\rangle$ with the number $m$, $n$ and $p$ denoting the photon number in mode $a$, $b$ and $c$, respectively. The system is truncated to two-photon excitation. And the Fock-state basis is shown in Fig. 1(a), where we consider the relation $\omega _b=2\omega _c$. To obtain the optimal analytic conditions for UPB, we analyse the interference paths shown in Fig. 1(b). There are so many Fock-state basis in Fig. 1(a), which lead to that it is difficult to calculate the optimal conditions for UPB. Here we only keep the Fock-state basis in Fig. 1(b), which are related to the interference paths. In this case, the the wave function of the system can be expressed as:

Considering the impact of the environment involved in the loss of the modes. We introduce the non-Hermitian Hamiltonian, which can be writen as

 

Fig. 1. (a) The Fock-state basis of the system. (b) The Fock-state basis, which are involved in the interference paths.

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Substituting the wave function Eq. (4) and non-Hermitian Hamiltonian Eq. (5) into the Schrödinger equation $i\partial _t|\psi \rangle =\widetilde {H}|\psi \rangle$, we can obtain the coupling equations for the coefficients,

By solving the coupling equation of the coefficients, we consider the steady state solution of this system

4. Comparison of the numerical results and analytic results

4.1 Numerical solution

The dynamics of the density matrix $\hat {\rho }$ of the system is governed by

In this paper, we only concern the zero-delay-time second-order correlation function in the steady state, so we need the steady-state density operator $\hat {\rho }_s$, which can be obtained by setting $\partial \hat {\rho }/\partial t=0$. The zero-delay-time second-order correlation function for mode $a$ is defined by

The second-order correlation function $g^{(2)}(0)<1$ corresponds to the sub-Poissonian statistics.

4.2 Comparison of numerical solutions and analytical conditions

In the following, the numerical results are provided to study the UPB effect by plotting the second-order correlation function $g^{(2)}(0)$ versus the system parameters. The Hilbert spaces of the system are truncated to five dimensions for cavity modes $a$, $b$ and $c$ respectively. And for convenience, we reset all parameters to units of dissipation rate $\kappa$.

First, we numerically study the UPB under the zero temperature ($\bar {n}_{th}=0$) and the results are compared with the optimal analytic condition shown in Eq. (8). In Fig. 2(a), we logarithmically plot $g^{(2)}(0)$ as a function of $g_1/\kappa$ and $\Delta _a/\kappa$ for mode $a$, and the other parameters are $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, and $F_{a}/\kappa =0.01$. According to the analytical results, we set the $E_1/\kappa =-\frac {1}{\varphi }\varepsilon$, where the parameters $\varphi =-648\kappa ^8-3882\kappa ^6\Delta ^2-7121\kappa ^4\Delta ^4-2542\kappa ^2\Delta ^6+2717\Delta ^8$ and $\varepsilon =356F\kappa ^6\Delta \lambda -1987\kappa ^4\Delta ^3\lambda -3606F\kappa ^2\Delta ^5\lambda -2115F\Delta ^7\lambda +8F\kappa ^2\Delta ^3\lambda ^3+18F\Delta ^5\lambda ^3$. The numerical results show that, the UPB can occur in this system. The valleys in $g^{(2)}(0)< 1$ correspond to the strong photon antibunching. The analytic condition is denoted by the white dotted line, which agrees well with numerical simulations. In Fig. 2(b), we logarithmically plot $g^{(2)}(0)$ as a function of $g_2/\kappa$ and $\Delta _a/\kappa$ for mode $a$, and the other parameters are $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.01$, and $E1/\kappa =\frac {1}{\varphi }\varepsilon$, which is consistent with the results in Fig. 2(a), the valleys in $g^{(2)}(0)< 1$ correspond to the strong photon antibunching and the analytic solution is denoted by the white dotted line, the analytical solutions and numerical simulations agree well in this situation. According to the results of Fig. 2(a) and Fig. 2(b), it is not difficult to find that the blockade regions is symmetrically distributed, which is in complete agreement with the results obtained by analytical calculation shown in Eqs. (8). Therefore, the method of preserving effective interference path adopted by us in dealing with this model is effective. In Fig. 2(c), we logarithmically plot $g^{(2)}(0)$ as a function of $E_1/\kappa$ and $\Delta _a/\kappa$ for mode $a$, where the shared parameters $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, and $F_{a}/\kappa =0.01$. According to the analytical results, we set the $g_1/\kappa =\frac {\lambda }{\sqrt {2}}$, and the parameters $\lambda =\sqrt {68\kappa ^2+\frac {18\kappa ^4}{\Delta ^2}+64\Delta ^2-\frac {\sqrt {(6\kappa ^2+13\Delta ^2)^2(9\kappa ^2+3\kappa ^2\Delta ^2+23\Delta ^4)}}{\Delta ^2}}$. Based on the results shown in the figure, the UPB can occur in this system, where the valleys in $g^{(2)}(0)< 1$ correspond to the strong photon antibunching, the shape of which is similar to the hyperbolic curve, and the white dotted lines represent analytic conditions. In Fig. 2(d), we logarithmically plot $g^{(2)}(0)$ as a function of $E2/\kappa$ and $\Delta _a/\kappa$ for mode $a$, where the shared parameters $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.01$ and $g2/\kappa =-\frac {\lambda }{\sqrt {2}}$. The valleys in $g^{(2)}(0)< 1$ correspond to the strong photon antibunching and the analytic solution is denoted by the white dotted line.In order to further compare and analyze the numerical and analytical results, we discuss the various coefficients appearing in the system model, and then determine the optimal coefficients setting for the UPB. In Fig. 3(a), we logarithmically plot $g^{(2)}(0)$ as a function of $\Delta _a/\kappa$, under different detuning $\Delta _a/\kappa$, where $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.1$, and $E/\kappa =\pm \frac {1}{\varphi }\varepsilon$. In the blue solid line we set the detuning $\Delta _a/\kappa =2$, in the black dotted line $\Delta _a/\kappa =3$, and in the red point line $\Delta _a/\kappa =4$. UPB occurs in all three curves, and each curve has two strong blockade valleys, which are basically match with the optimal analytic conditions. With the increasing of the parameter $\Delta _a/\kappa$, the intensity of UPB increases gradually. At the same time, the absolute value of the second-order nonlinear coefficient $g/\kappa$ also increase. Therefore, we cannot determine that a larger $\Delta _a/\kappa$ value is favorable for this system to implement UPB. Next, we choose $\Delta _a/\kappa =3$ for further analysis of the other coefficients in the system.

 

Fig. 2. (a) Logarithmic plot of zero-delay-time second-order correlation functions $g^{(2)}(0)$ as a function of $g_1/\kappa$ and $\Delta _a/\kappa$ for mode $a$, where the shared parameters $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.01$ and $E_1/\kappa =-\frac {1}{\varphi }\varepsilon$. (b) Logarithmic plot of $g^{(2)}(0)$ as a function of $g_2/\kappa$ and $\Delta _a/\kappa$ for mode $a$, where the shared parameters $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.01$ and $E_2/\kappa =\frac {1}{\varphi }\varepsilon$. (c) Logarithmic plot of $g^{(2)}(0)$ as a function of $E1/\kappa$ and $\Delta _a/\kappa$ for $\Delta _b=\Delta _a/4$, $\Delta _c=3\Delta _a/4$, $F_{a}/\kappa =0.01$ and $g_1/\kappa =-\frac {\lambda }{\sqrt {2}}$. (d) Logarithmic plot of $g^{(2)}(0)$ as a function of $E2/\kappa$ and $\Delta _a/\kappa$ for $\Delta _b=\Delta _a/4$, $\Delta _c=3\Delta _a/4$, $F_{a}/\kappa =0.01$ and $g_2/\kappa =\frac {\lambda }{\sqrt {2}}$.

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Fig. 3. (a) Under different detunings $\Delta _a/\kappa$, we logarithmically plot of $g^{(2)}(0)$ vs the second-order nonlinear interaction strength $g/\kappa$, with $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.1$ and $E/\kappa =\pm \frac {1}{\varphi }\varepsilon$ (b) Under different weak drive factor $F_{a}/\kappa$, we logarithmically plot of $g^{(2)}(0)$ vs the nondegenerate parametric amplification factor. $E/\kappa$, with $\Delta _a/\kappa =3$, $\Delta _b=\Delta _a/8$, $\Delta _c=7\Delta _a/8$ and $g/\kappa =\pm \frac {\lambda }{\sqrt {2}}$.

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In Fig. 3(b), we logarithmically plot $g^{(2)}(0)$ as a function of $E/\kappa$ under different $F_a/\kappa$, where $\Delta _a/\kappa =3$, $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$ and $g/\kappa =\pm \frac {\lambda }{\sqrt {2}}$. In the blue solid line we set the detuning $F_a/\kappa =0.05$, in the black dotted line $F_a/\kappa =0.08$ and in the red point line $F_a/\kappa =0.1$. The UPB effect occurs on all curves, and each curve has two dips, which corresponds to the analytic conditions. With the increase of the driving strength $F_a/\kappa$, the UPB effect of the system became weaker and weaker. Meanwhile, the absolute value of the non-degenerate parametric amplification coefficient $E/\kappa$ also moves in the direction of the increase.

Finally, we investigate the UPB effect under nonzero temperatures. In Fig. 4, we plot $g^{(2)}(0)$ vs the second-order nonlinear interaction strength $g/\kappa$ with $\Delta _a/\kappa =3$, $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.05$, and $E/\kappa =0.05$. In the black solid line we set $\bar {n}_{th}=0$, in the red dotted line $\bar {n}_{th}=0.01$, and in the blue point line $\bar {n}_{th}=0.02$. The results show that the strongest UPB point appears at $g/\kappa =-2.32$, just as predicted by analytical conditions. Moreover, when $\bar {n}_{th}$ changes in a large ranges, the UPB does not change significantly, which indicates that this scheme is not sensitive to the change of the reservoir temperature.

 

Fig. 4. Under different thermal photons $\bar {n}_{th}$, we logarithmically plot of the $g^{(2)}(0)$ vs he second-order nonlinear interaction strength $g/\kappa$, with $\Delta _a/\kappa =3$, $\Delta _b=\Delta _a/5$, $\Delta _c=4\Delta _a/5$, $F_{a}/\kappa =0.05$ and $E/\kappa =0.05$.

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5. Conclusions

In summary, we have investigated the unconventional photon blockade in a three-wave-mixing system, where the three-wave-mixing mediates can convert a photon with high frequency into two photons with different low frequencies. We simplify the complex wave function composition of the system by preserving the state on the effective interference paths. By analytic calculation, we obtain the optimal condition for strong antibunching, and the discussions of the optimal conditions are presented. Through numerical calculation, we find that the system can realize UPB in the high-frequency mode. According to comparisons of analytical conditions and numerical calculations, ther are in good agreement. The results show that the method of preserving effective interference paths is effective in dealing with this complex model to achieve the UPB effect. So, this method may be useful for dealing with other complex models, and we will discuss it further in the following research. The coefficients in the system are discussed in detail, and the UPB immunes to reservoir temperature.